Subset of Locally Finite Set of Subsets is Locally Finite

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $\FF \subseteq \powerset S$ be a set of subsets of $S$.

Let $\GG \subseteq \FF$.

If $\FF$ is locally finite then $\GG$ is locally finite.

Proof

We prove the contrapositive statement:

If $\GG$ is not locally finite then $\FF$ is not locally finite.

Let $\GG$ not be locally finite.

By definition of locally finite:

$\exists x \in S : \forall N \subseteq S : N$ is a neighborhood of $x : N$ intersects an infinite number of sets in $\GG$.

By definition of subset:

$\forall X \in \GG : X \in \FF$

Hence:

$\exists x \in S : \forall N \subseteq S : N$ is a neighborhood of $x : N$ intersects an infinite number of sets in $\FF$.

By definition, $\FF$ is not locally finite.

The result follows from Rule of Transposition.

$\blacksquare$